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Warning 5.4.6.7. The collection of minimal $\infty$-categories has poor closure properties:

• If $\operatorname{\mathcal{C}}$ is a minimal $\infty$-category and $K$ is a simplicial set, then the $\infty$-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ need not be minimal (even in the case $K = \Delta ^1$).

• If $\operatorname{\mathcal{C}}$ is a minimal $\infty$-category and $q: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram, then the $\infty$-categories $\operatorname{\mathcal{C}}_{/q}$ and $\operatorname{\mathcal{C}}_{q/}$ need not be minimal (even in the case $K = \Delta ^0$).

• If $\operatorname{\mathcal{C}}$ is a minimal $\infty$-category and $\operatorname{\mathcal{D}}$ is equivalent to $\operatorname{\mathcal{C}}$, then $\operatorname{\mathcal{D}}$ need not be minimal.